The insights in the main paper regarding "twistedness"
reflect an intuitive understanding of complexity which calls for deeper insight
to understand how twistedness works and why it may be vitally important in some
psycho-social processes -- as well as being highly problematic in others. Part
of the difficulty in approaching this matter is that "twistedness"
is in most cases used unthinkingly as a pejorative term to characterize a pattern
which is felt to inhibit right-thinking and clarity. The argument here is that,
given its importance at every scale in nature, from the organization of nebula
to the organization of the human cell, there is a case for distinguishing various
forms of twistedness and understanding their function. This could be especially
valuable to reconciling apparently irreconcilable understandings in society.

The merit of focusing on the nature and function of twisting in
DNA is that it provides a rich natural template. It offers a sense of the degree
of complexity that it may be required to master in order to comprehend how twistedness
"works" in practice. It might also be argued that, as a process active
in every human body and inherent to human life, humans may well have some kind
of profound intuitive understanding of how it works and the "rightness"
of such working. Some of the very explicit dynamics of this process may also
offer patterns for understanding how the inhibiting effects of "twistedness"
may be addressed when they are perceived to be a constraint on human development.

Chromosomal DNA molecules are very long and thin. There is over
a metre of DNA in every human cell in a space of some 0.0006 centimetres diametre.
If DNA were constrained to be linear it would not fit into a cell. It must therefore
fold many times to fit within the confines of a cell. The DNA is composed of
10** base pairs. This density of packing results in tangles and knots in the
DNA that are essential to enable the cell to divide (involving transcription
and replication).

Structure of DNA

DNA is a double stranded molecule composed of two polarized strands
(of deoxyribonucleotide polymers) which run in opposite directions (termed antiparallel)
and wind around a central, common axis -- one is entwined about the other such
that an overall helical shape results (known as a plectonemic helix). Both are
wound in a right-handed manner. This structure is to be contrasted with a paranemic
helix, in which a pair of coils lie side by side without interwinding. The strands
are occasionally distinguished as the Watson strand and the Crick strand.

In the case of the molecular structure of eukaryotic chromosomes in each human
cel, 2 meztres of DNA is packaged into the cell nucleus. To access the information,
it must be unwound as a double helix and needs to be "spread out" in the nucleus.
However during cell division (mitosis), in order to move them around, they are
packaged as follows into dense bundles:

Supercoiling (coil of solenoids is itself coiled): coiled coil is then folded
- as in mitotic chromosome, namely a 10,000 fold reduction in length

Each nucleotide base of one strand is paired with a nucleotide
base on the other strand to create a stable structure of the two polymers. The
pairing of the four types of bases (A, T, C, G) by hydrogen bonds is not random:
an A pairs with a T and a G pairs with a C. The bases on the outside of the
helix are exposed to solvent within two grooves along the helix, the "major
groove" and the "minor groove". It is within these grooves that DNA interacts
with other molecules. The three structural variation of these grooves ("A",
"B" and "Z" DNA), which differ in the relationship between the bases and the
helical axis, offer one mechanism by which reactivity of DNA is modulated:

B-DNA : Fully hydrated DNA, the most common encountered in vivo. Owing to
the location of the helical axis in the center of the base pairs, the edges
of the base pairs are about equally deep in the interior.

A-DNA : When B-DNA is dehydrated, there is a reversible structural change
to A-DNA

Z-DNA : Unlike B-DNA and A-DNA, Z-DNA is a left-handed helix. The conformational
change from B-DNA to Z-DNA is one mechanism for relief of the torsional strain
found in B-DNA in vivo, and may serve as a switch mechanism to regulate gene
expression.

In circular double helix DNA (closed circular ccDNA), both strands
are covalently joined to form a circular duplex molecule. The geometry of such
an assembly is such that its number of coils cannot be changed without first
breaking one of its strands. This topological "dilemma" is resolved
within the cell -- to ensure proper biological functioning -- by specialized
enzymes that unknot, untwist and unwind the DNA to enable replication and then
reform the compact mode thereafter.

Heptad repeats: The coiled coil is a ubiquitous protein-folding
motif. The accepted hallmark of the coiled coil is the seven-residue heptad
repeat..A coiled-coil protein consists of two identical strands of amino acid
sequences that wrap around each other. The amino acids in a coiled-coil structure
reside on seven different structural positions on the coil, forming a heptad
repeat (see The Heptad
Repeat of The Coiled-coil Structure). Heptad repeats are characteristic
of certain proteins. (see also images of David Gossard. Coiled
Coils. 2003). Most coiled-coil sequences contain heptad repeats, namely
seven residue patterns -- denoted abcdefg -- in which the a and d residues (core
positions) are generally hydrophobic. As there are 3.6 residues to each turn
of the alpha-helix, these a and d residues form a hydrophobic seam, which, as
each heptad is slightly under two turns, slowly twists around the helix. The
coiled-coil is formed by component helices coming together to bury their hydrophobic
seams. As the hydrophobic seams twist around each helix, so the helices also
twist to coil around each other, burying the hydrophobic seams and forming a
supercoil. It is the characteristic interdigitation of side chains between neighbouring
helices, known as knobs-into-holes packing, that defines the structure as a
coiled coil (see Jenny Shipway. An
Introduction to Coiled Coils. 2000) [more
| more].

Forms of DNA

Supercoiled (or "knotted"): Double stranded
circular (or linear) DNA can have tertiary or higher order structure. Superhelicity
is therefore sometimes referred to as DNA's tertiary structure. Supercoils
refer to the DNA structure in which double-stranded circular DNA twists around
each other. This is termed supercoiling, supertwisting or superhelicity --
meaning the coiling of a coil, also understood in terms of knots. Only topological
closed domains (such as a covalently closed circle) can undergo supercoiling.
A linear molecule can have topological domains as long as there is a region
of the DNA bounded by constraints on the rotation of the DNA double helix.
Eukaryotic DNAs in association with nuclear proteins acquire superhelical
conformation in chromosomes.

Adding a twist to the DNA (as catalyzed by an enzyme), imposes a strain. A
DNA segment so strained that is closed into a circle would then contort into
a figure of eight (or its topological equivalents) -- the simplest supercoil.
This is the shape that a circular DNA assumes to accomodate one too many or
one too few helical twists. For each additional helical twist that is accomodated,
the lobes will show one more roation about their axis. Such superhelicity
results in more compact structures. In any other naturally found geometry,
the DNA is either under- or overwound. Its helical axis does not lie in a
plane or on the surface of a sphere because of writhing and twisting of it.
This is the physical solution to the potential (torsional) energy minimization
problem. Supercoiling can therefore be :

negative (right-handed): Supercoils formed by deficit in link
are called negative supercoils. They result from underwinding, unwinding
or subtractive twisting of the DNA helix (due to a deficit in link). The
two lobes of the figure of eight then appear rotated counterclockwise
with respect to each other. All naturally occuring double stranded DNAs
are negatively supercoiled. Negative supercoiling facilitates DNA-strand
separation during replication, recombination and transcription. All the
naturally occuring double stranded DNAs are negatively supercoiled (including
bacterial and viral circular duplex DNAs).

positive (left-handed): Supercoils formed by an increase in link
are called positive supercoils. They result from tighter winding or overwinding
of the DNA helix (due to an increase in link) resulting in extra helical
twists. The two lobes of the figure of eight then appear rotated clockwise
with respect to each other. This would compact DNA as effectively as negative
supercoiling, but would make strand separation much more difficult.

In non-dividing eukaryotic cells, chromosomal DNA is wrapped around
a nucleosome core which consists of highly basic proteins called histones.
The DNA is wrapped around the nucleosome in a left-handed solenoidal arrangement.
This negative supercoiling is one of the forms taken up by underwound
DNA.

Relaxed: Circular DNA without any superhelical
twist is known as a relaxed molecule. DNA in its relaxed (ideal) state usually
assumes the B configuration. In a relaxed double-helical segment of DNA, the
two strands twist around the helical axis once every 10.6 base pairs of sequence.
Relaxed, closed circular DNA, is defined as DNA which has no supercoils when
constrained to lie flat in a plan. The following structures are consistent
with the relaxed state: (a) Linear DNA (either straight or curved) (b) Closed
circular DNA, provided its axis lies in a plane or on the surface of a sphere

Supercoiling is thus vital to two major functions. It helps pack
large circular rings of DNA into a small space by making the rings highly compact.
It also helps in the unwinding of DNA required for its replication and transcription.
Supercoiled DNA is thus the biological active form. The normal biological functioning
of DNA occurs only if it is in the proper topological state.

Descriptive properties associated with supercoiling

"Supercoiling" is an abstract mathematical property, and represents
the sum of what are termed "twist" and "writhe". "Supercoil"
is seldom used as a noun with reference to DNA topology. It is the combination
of twists and writhes that impart the supercoiling, and these occur in response
to a change in the linking number. A coiled structure is at a higher energy
(less stable). When the linking number is reduced in closed circular DNA, the
molecule supercoils by minimizing twisting and bending. To partially relieve
the strain introduced by the change in linking number (a 'deficit' in the link),
the DNA must distort in other ways -- compensating with a change in twist or
writhe. These are, physically, the two ways that the DNA can do so. The relationship
of twist, writhe and supercoiling is expressed by the equation S = T + W (known
as White's formula). Twist and writhe are geometric quantities. Unusually, link
as a topological property is equal to the sum of two geometric properties. Their
values change if the ribbon is deformed in space. Link, twist and writhe can
be either positive or negative. Link is always an integer, whereas twist and
writhe can take any real values.

Writhing: Global contortions of circular DNA are
described as "writhe". The writhing number describes the supertwisting
or supercoiling of the helix in space. It is the number of turns that the
duplex axis makes about the superhelix axis. Writhe describes the supercoiling,
the coiling of the DNA coil. It is a measure of the DNA's superhelicity (supercoiling)
and can be positive or negative. Writhe is a measure of the coiling, bending
or non-planarity of the axis of the double helix. A right-handed coil is assigned
a negative number (negative supercoiling) and a left-handed coil is assigned
a positive number (positive supercoiling). When a molecule is relaxed and
contains no supercoils, the linking number = the twist number since W= 0 The
linking number of relaxed DNA is L 0 L 0 = N/10.5, where N is the number of
base pairs in the DNA fragment.

Twisting: Twist is the number of helical turns
in the DNA, i.e., the complete revolutions that one polynucleotide strand
makes about the duplex axis in the particular conformation under consideration.
Twist is normally the number of base pairs divided by 10.4, that is the number
of bases per turn of the helix. Twist is altered by deformation and is a local
phenomenon. The total twist is the sum of all of the local twists. Twist is
a measure of deformation due to a twisting motion.
Twist and writhe are interconvertable. In part because chromosomes may be
very large, segments in the middle may act as if their ends are anchored.
As a result, they may be unable to distribute excess twist to the rest of
the chromosome or to absorb twist to recover from underwinding -- the segments
may become supercoiled, in other words. In response to supercoiling, they
will assume an amount of writhe, just as if their ends were joined.

Linking number: This is a topological property
that determines the degree of supercoiling; It defines the number of times
a strand of DNA winds in the right-handed direction around the helix axis
when the axis is constrained to lie in a plane. It is the number of times
that one DNA strand crosses about the other when the DNA is made to lie flat
on a plane. If both strands are covalently intact, the linking number cannot
change. Link is thus a topological invariant, remaining unaltered even if
the two curves are deformed in space -- as long as neither is cut. Topology
theory indicates that the sum of T and W equals to linking number: L=T+W.
For example, in the circular DNA of 5400 basepairs, the linking number is
5400/10=540. When a molecule is relaxed and contains no supercoils, the linking
number = the twist number since W= 0. Thus if there is no supercoiling, then
W=0, T=L=540. If there is positive supercoiling, W=+20, T=L-W=520. In the
special cases in which axis of the double helix remains in a plane or on the
surface of a sphere, then twist equals the linking number, and there is no
writhe, but all other cases are considerably more complex. Supercoiling can
be caused even by an increase in the linking number (though this does not
occur in nature).

Density: the density of supercoiling.is useful
to define as a property that distinguishes DNAs varying significantly in size.
Superhelical density is the number of supercoils per turn of helix. It is
denoted by the Greek letter sigma. It is defined as the number of turns that
have been added or subtracted in the supercoiled DNA, compared to the relaxed
state, divided by the total number of turns in the DNA if it were relaxed
(which would normally be bp/10.5). Typically, sigma is between -.05 and -.07
(5-7% underwinding) in isolated natural DNA

Link altering enzymes: The functionality of DNA
is related to its topology which is maintained by enzymes that are capable
of altering it. Nature has come up with particular enzymes that control the
knottedness (as well as other topological states such as twist-induced supercoiling)
of DNA. The exact ability of these enzymes to locate a knot in a circular
DNA is an unresolved question in molecular biology. Known as Topoisomerases,
these enzymes change the structure by altering the DNA link of a molecule.
This is achieved by temporarily breaking one of the strands, passing the other
strand through it, and then resealing the bonds. This effectively changes
the linking number in the DNA. The enzymes are of two types:

Type-1: function by creating transient single-strand breaks in DNA,
altering the link by one, by cutting one strand and passing the other
strand through the break.

Type-2: alter the link by two, by breaking both the strands of the double
helix at the same time and passing a segment of the double helix through
the break.

Many topoisomerase enzymes sense supercoiling and either generate or dissipate
it as they change DNA topology.

Replication: Level of supercoiling is known to
be important for initiation of replication. In DNA replication, the two strands
of DNA have to be separated, which leads either to overwinding of surrounding
regions of DNA or to supercoiling. During replication, only part of the DNA
unwinds (200 bps) while the rest of the DNA still remain configured at 10
bps per turn. A specialized set of enzymes (gyrase, topoisomerases) is present
to introduce supercoils that favor strand separation; The degree of supercoils
can be quantitatively described. Because of the wound configuration of DNA,
biochemical transactions requiring strand separation necessitate chromosome
movement (spin) about the long axis of the DNA. DNA replication, recombination
and transcription all require DNA rotation. During DNA synthesis the rotation
speed approaches 6000 rpm.

DNA Replication, RNA transcription, and Gene expression: Negative
supercoiling in cells is energetically unfavorable and must be introduced
in some manner: Therefore transcription is supercoil dependent. Topological
domains may thus alter local regulation of gene expression. The overall level
of supercoiling in a cell could have a global effect on gene expression. DNA
is transcribed into mRNA in the nucleus. There are particular codons to which
the enzymes are begin the transcription. The mRNA travels to, and attaches
itself to a ribosome. There, the nucleotides are read from the start codon,
in three letter sections, each of which code for a particular amino acid (some
amino acids have multiple codons that code for them). The tRNA brings amino
acids from the surrounding cytosol to the ribosome, and attach them in the
order coded by the mRNA. The amino acids are then bonded together by peptide
bonds. They are in a long linear chain, which is the primary conformation.
The conformation to secondary structure is usually spontaneous, based on the
interactions between the amino acids within the polypeptide. However, when
going to the tertiary and quarternary conformations, there are usually larger
proteins called chaperones that assist in the folding.

Denaturation, melting, breathing and unzipping:
A physical property cricual to the function of DNA in replication and trqnscription
is the ease with which its component parts can separate and be rejoined.
This process is sometimes referred to as "melting" and "reannealing",
or "denaturation" and "renaturation". DNA denaturation
(or "melting") is due to the breakage of the hydrogen bonds in
the Watson-Crick base-pairs, and is therefore reversible. This "unzipping"
can be brought about by several processes. Under physiological conditions,
local DNA-breathing occurs spontaneously due to thermal fluctuations. This
opens up transient bubbles of a few tens of base pairs. These breathing
fluctuations may be supported by single-strand binding proteins, thereby
lowering the DNA base pair stability. DNA breathing and the lability of
local stretches of the DNA double-helix is essential for numerous physiological
processes such as the association of single-strand binding proteins, and
the initiation of replication and transcription. (see Ralf Metzler and Andreas
Hanke. Knots,
bubbles, untying, and breathing: probing the topology of DNA and other biomolecules.
2004). [more
| more]

DNA-knots: DNA knots can arise in various biological
reactions involving circular DNA molecules. Site specific recombination enzymes
are known to produce specific families of knots. Identification of the knot
types indicates the mechanism of action of a given enzyme and the overall
shape of supercoiled circular DNA molecules at the moment of knotting [more
| more].
Internal pairing in short single-stranded DNA can be utilised for the construction
of different types of DNA knots [images].
The presence of knots inhibits the assembly of chromatin. Knotted chromosomes
cannot be separated during mitosis, and knots in a chromosome may serve as
topological barriers between different sections of chromosomes, such that
the genomic structural organisation is altered.

Energy associated with different structures

The energy of the molecule changes if there is a change in pitch
(that is, the number of bases per full turn) or bending of the double helix
ring. Even a small change in the pitch of the DNA results in a large increase
in energy

Minimum energy: Linear DNA assumes the B configuration
because it is the one of minimum energy. Linear molecules of DNA assume a
configuration known as the b-configuration. Deviation from this relaxed state
increases the energy of the DNA molecule, although circular DNA of large diameter
increases it least.

Higher energy: In the ring form too, the DNA double
helix tries to attain the state of minimum energy. The DNA ring approximates
the b-configuration of the linear molecule while trying to attain the state
of minimum energy. This packaging of DNA deforms it physically, thereby increasing
its energy. Such an increase in stored (potential) energy within the molecule
is then available to drive reactions such as the unwinding events that occur
during DNA replication and transcription. Too much stored energy is not necessarily
a good thing, though. In nature, this problem is addressed by having DNA form
supercoils, in which the helical axis of the DNA curves itself into a coil.
Supercoiling or the formation of a superhelix structure minimizes the excess
energy that builds up when DNA molecules are deformed during the packing process.

At this point, it's a good idea to mention that supercoiling is not necessarily
the only solution to the problem of normalizing the number of base pairs per
helix in an unwound piece of DNA. You could also separate the two strands
by breaking the hydrogen bonds between complementary bases in contiguous base
pairs until the remaining DNA has the correct number of base per per turn.
In terms of energy needed, though, it requires a lot more energy to break
the H-bonds than to supercoil. Nevertheless, strand separation does occur
during replication and transcription and it turns out that it is the physics
of the underwinding that facilitates the strand separation. Cruciform structures
also require some unpairing of the base pairs and, again, it is the underwinding
that maintains the required strand separation.

References

Ralf Metzler and Andreas Hanke. Knots, bubbles, untying, and breathing:
probing the topology of DNA and other biomolecules. 2004 [text]